1.1 Vectors and scalars
When people askâWhat is a vector?â it is as difficult to answer asâWhat is a number?â Both vectors and numbers are abstract ideas which represent more concrete quantities. We start by learning that two apples added to two apples gives us four apples, two pencils added to two pencils gives us four pencils, and so on using physical objects, and it is some time before we link this with the more abstract concept 2 + 2 = 4. With a vector there are two quantities involved in the representation, and we generally think of these as magnitude and direction, and we often use the term length as an alternative for magnitude. So a vector is defined as something having both magnitude and direction, and anything which has just a magnitude attached to it is called a scalar. In this book all our scalars will be real numbers, but readers should be aware that there are vector spaces for which the scalars are complex numbers or even more exotic beings.
One way of differentiating between vectors and scalars is by considering the difference between the distance between two points, which is a scalar, and the displacement of one point from another, which is a vector, and which we can regard as what we have to do to get from one point to another. In this case we need to know not only how far we have to go, but also in which direction. Buckingham Palace is 1.25 km from Trafalgar Square, but if someone is starting from Trafalgar Square and wishes to get to Buckingham Palace, it is no good walking 1.25 km to the east!
Examples of scalar quantities are distances, speeds and masses, and examples of vector quantities are displacements, velocities, weights.
Notation
We shall represent vectors in bold type and scalars will be written in italics, so v represents a vector, but s represents a scalar. The vector which represents displacement from a point A to a point B will be written as AB, and the vector from B to A as BA. This indicates how vital it is to make the direction clear on a, diagram, and we shall use arrows to indicate direction, as in Fig 1.1.